Yup.
I'm an only child, so I provided all the support for my parents and eventually moved Mum in with me. It's hard.
I also became my DH's carer. His children were conspicuous by their absence, though they purported to care. I recall my husband warning me that he didn't think that the kids would come to his funeral (They're an hour's flight away.)
I told him that of course they'd come. I didn't realise until too late that he meant that they wouldn't be there to support me.
Each time their mother acquired a new partner, they were supportive of that - presumably because it saved them from having to step up. (She currently lives with a chap who has two adult daughters who moved in with him.)
OP, that type of person is always very happy to keep their parents out of a care home, provided someone else is providing the care.
You have a couple of choices that I can see: buy the home that you're talking about and tell the siblings to step up; stay with your mum and warn the siblings that they need to step up, including providing respite for you. (Spoiler: they won't.)
When my husband's ex lost her Affair Partner, DH's SIL actually complained to us because the AP left his entire estate to the ex and nothing to the grandchild. That confirmed my suspicion that the main concern was money.
After DH died, he was proved right. The kids didn't attend the funeral, but watched online. (To be fair, it was during one of the lockdowns, but the by then adult grandchild and her mother could definitely have attended.)
After years of biting my tongue, I lost it a month or so after the funeral. The last communication that I had was indirect. The daughter and grandchild phoned the solicitor to say that their cheques had been refused by the bank. (I got it sorted out - the bank thought that I was being scammed.)
I'm merailing here, aren't I? The point I'm trying to make, OP is that your experience has shown you that your siblings are only interested in themselves and in pretending to be supportive of your mother. They won't change.
What happens if you become unwell? What do your siblings intend to have as back-up? Do they expect your children to be carers?
When I fell ill, I phoned DH's daughter to explain that her dad would need care if anything happened to him. The response was "You're not thinking of leaving him are you?"
When I got the all clear I phoned and reiterated that if anything happened, her father would need care. The response: "But you're all right, aren't you?"
Either go ahead and buy that house, OP or give your siblings warning that they need to step up...but I'm betting that they won't.